The Symmetry Groupoid and
Weighted Signature of a Geometric Object
Peter J. Olver†
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
[email protected]
http://www.math.umn.edu/∼olver
Abstract. We refine the concept of the symmetry group of a geometric object through
its symmetry groupoid, which incorporates both global and local symmetries in a common
framework. The symmetry groupoid is related to the weighted differential invariant signature of a submanifold, that is introduced to capture its fine grain equivalence and symmetry
properties. Applications to the recognition and symmetry properties of digital images are
indicated.
1.
Introduction.
Roughly speaking, the “symmetry group” of a geometric object is the set of transformations that leave the object unchanged. The fact that symmetries form a group follows
from the fact that the composition of any two symmetries is again a symmetry, as is the
inverse of any symmetry.
However, to be precise, one must, a priori, specify the overall class of allowed transformations G to which the symmetries are required to belong. In Klein geometries, [15],
G represents a finite-dimensional Lie transformation group, or, slightly more generally,
a local Lie group action, [28], the most familiar example (of the former) being the Euclidean group of rigid motions of Euclidean space. Other well-studied examples are the
geometrically-based (local) Lie transformation groups consisting of affine transformations,
of projective transformations, or of conformal transformations relative to some, possibly indefinite, metric, the last example requiring that the dimension of the space be ≥ 3.
†
Supported in part by NSF Grant DMS 11–08894.
August 19, 2015
1
More generally, the class G could be a pseudo-group of transformations, [8, 10, 11, 21, 35],
for example the pseudo-group of all local homeomorphisms, of local diffeomorphisms, of
volume-preserving local diffeomorphisms, the pseudo-group of canonical transformations
of a symplectic or Poisson manifold, or the pseudo-group of conformal transformations
of the Euclidean or Minkowski plane. Since finite-dimensional local and global Lie group
actions can be viewed as Lie pseudo-groups of finite type, it is tempting to develop our
ideas in the latter more general, but more technically demanding, framework. However, to
keep the exposition here at a reasonable level, we shall restrict ourselves to the Lie group
context, leaving extensions of these results to infinite-dimensional pseudo-group actions as
projects for the interested reader.
The specification of the underlying transformation group G is important, because
the symmetry group associated to an object depends upon it. For example, suppose
S = ∂Q ⊂ R 2 is the boundary of the square Q = {−1 ≤ x, y ≤ 1}. If G = SE(2) is the
special Euclidean group consisting of orientation-preserving rigid motions — translations
and rotations — then the symmetry group of S is the finite group Z4 = Z/4 Z ⊂ SE(2)
consisting of rotations by multiples of 90◦ around the origin. Expanding to G = E(2),
the full Euclidean group containing translations, rotations, and reflections, produces an 8
element symmetry group Z2 ⋉ Z4 ⊂ E(2) containing the above rotations along with the
four reflections across the two coordinate axes and the two diagonals. On the other hand, if
G = H(R 2 ) is the pseudo-group of all local homeomorphisms, then the induced symmetry
group of S is considerably larger. For example, one can easily construct a homeomorphism
ψ: R 2 → R 2 that maps S to the unit circle C = {x2 + y 2 = 1}. Then the composition
ψ −1 ◦ R ◦ ψ, in which R is any rotation around the origin, defines a homeomorphism symmetry of S. Clearly, in this much broader context, the square now possesses an infinite
number of symmetries. One can clearly produce a similar example in the smooth category, that is, the pseudo-group of local diffeomorphisms of the plane, by symmetrically
smoothing out the corners of the square.
Let us, from here on, fix the underlying Lie transformation group G acting on the space
M , and require symmetries of objects S ⊂ M to be elements thereof. As emphasized by
Weinstein, [39], the usual characterization of the symmetry group fails to capture all the
symmetry properties of common geometric objects, and its full set of symmetries forms a
more general object, known as a groupoid , [25]. A good example is a square tiling of the
plane. If the tiling is of infinite extent, then its Euclidean symmetry group is an infinite
discrete group generated by translations in directions of the two tiling axes, combined
with the rotation and reflection symmetries of each underlying square tile. However, a
bounded portion of the square tiling, e.g. a bathroom floor as in Figure 1, has no translation
symmetries and possibly, depending on the shape of the outline of the portion, no rotation
or reflection symmetries either, even though to the “untrained” eye a tiled floor remains
highly symmetric. One productive way to mathematically retain our aesthetic notion of
symmetry is to introduce the concept of a “local symmetry” of a geometric object. Local
symmetries no longer form a group, but, rather, have the structure of a groupoid, which
we call the symmetry groupoid . Indeed, in his survey article, Brown, [5], emphasizes that
groupoids form the appropriate framework for studying objects with variable symmetry; a
relevant example is a surface, part of which is spherical, of constant curvature and hence
2
Figure 1.
A Bounded Tiling.
a high degree of local symmetry, another part is contained in a surface of revolution, and
hence admit a one-parameter family of local symmetries, while other parts have variable
principal curvatures, and hence at most a discrete set of local symmetries. While the
overall surface may have trivial global symmetry group, its symmetry groupoid retains the
inherent local symmetries of its component parts. In Section 2, we present our version of
this basic construction, illustrated by several examples.
The key distinction between groups and groupoids is that one is only allowed to multiply elements of the latter under certain conditions. Historically, groupoids were first
introduced in the 1920’s by Brandt, [3], to study quadratic forms. More directly relevant to our concerns is Ehresmann’s fundamental paper, [10], in which he introduced the
powerful tools of Lie groupoids and jet bundles order to study the geometric properties
of partial differential equations and, in particular, Lie pseudo-groups. Here, the fundamental examples are the groupoids consisting of Taylor polynomials or series, a.k.a. jets
of diffeomorphisms belonging to the pseudo-group, [11, 35]. Observe that one can only
algebraically compose a pair of Taylor polynomials, p2 ◦ p1 , if the source of p2 , meaning its
base point, matches the target of p1 , meaning its image point.
While the symmetry groupoid construction developed in Section 2 applies to very
general geometric objects, our primarily focus is on (sufficiently smooth) submanifolds. In
the equivalence method of Élie Cartan, [9, 14, 29], the functional relationships or syzygies
among its differential invariants serve to prescribe the local equivalence and symmetry
properties of a sufficiently regular submanifold. Motivated by a range of applications in
image processing, one employs a suitable collection of differential invariants to parametrize
the differential invariant signature of a submanifold, [6]. The second topic of this paper is
to understand how the differential invariant signature is related to the symmetry groupoid
of the underlying submanifold. In particular, the codimension and index of the signature
directly correspond to the dimension and cardinality of the symmetry groupoid at the
corresponding point. In particular, the cogwheel curves of [27], which appear to violate
the signature symmetry conditions, are explained correctly in the context of their local
3
symmetry groupoids. Motivated by other examples appearing in [27], and in response,
the extended curve signatures proposed in [17] — which was then applied to develop
remarkably effective algorithms for the automatic assembly of apictorial jigsaw puzzles,
[18] — we shall also investigate submanifolds of variable signature rank. The goal is to
determine to what extent the equivalence method of Cartan can be extended to this more
general context. To this end, we introduce a weighted version of the differential invariant
signature, based on uniformly sampling the submanifold relative to some group-invariant
measure. We then show how the weighted signature might be employed to analyze the
equivalence and symmetry groupoid properties for a broad range of submanifolds. In our
analysis of the weighted signature, we are led to a group-invariant version of the celebrated
coarea formula of geometric measure theory, [12, 26], that is of independent interest; see
Section 4 for details. Applications in image processing and elsewhere will be the subject
of future investigations.
2.
The Symmetry Groupoid of a Geometric Object.
In this section, we shall formalize the notion of the symmetry groupoid of a geometric
object, and explain how it better captures the inherent symmetry properties than the more
traditional, coarser symmetry group. For simplicity, we shall assume that G is a finitedimensional Lie group acting smoothly and globally (but not necessarily transitively) on
a manifold M . As noted above, with some care, one can readily extend the following
constructions to local Lie group actions as well as to infinite-dimensional Lie pseudogroups.
Let S ⊂ M be a subset. For us, the most important case is when S is a submanifold,
but for the time being we allow S to be arbitrary. The most common definition of a
symmetry of the subset S is a group transformation g ∈ G that preserves S, meaning that
g · S = { g · z | z ∈ S } = S.
(2.1)
The set of such symmetries is easily shown to form a subgroup
GS = { g ∈ G | g · S = S } ⊂ G,
(2.2)
which we call the global symmetry group of S ⊂ M . Its elements will be referred to as
global symmetries from here on.
As noted in the introduction, global symmetries may fail to fully capture the more
subtle symmetry properties of objects, and thus we now refine the concept by suitably
localizing the symmetry requirement.
Definition 2.1. A group transformation g ∈ G is a local symmetry of S based at the
point z ∈ S if there is an open neighborhood z ∈ U ⊂ M such that
g · (S ∩ U ) = S ∩ (g · U ).
(2.3)
We denote the set of local symmetries based at z by Gz ⊂ G. Note that, at the very least,
the identity e ∈ Gz .
4
Any global symmetry of S is clearly a local symmetry at each z ∈ S, and so GS ⊂ Gz .
Moreover, the set of global symmetries can be identified as the intersection of all local
symmetry sets:
\
GS =
Gz .
z∈S
Although the set of all local symmetries does not, in general, form a group — see below
for examples — it does form a groupoid, in accordance with the following definition, [25].
Definition 2.2. A groupoid over a base S is a set G along with a pair of surjective
maps σ, τ : G → S, called the source and target maps, a binary operation (α, β) 7−→ α · β,
called multiplication, that is defined on the set
G ⋆ G = { (α, β) | σ(α) = τ (β) } ⊂ G × G,
and an injective identity map ε : S → G, satisfying the following conditions:
• Source and target of products: σ(α · β) = σ(β), τ (α · β) = τ (α), for (α, β) ∈ G ⋆ G.
• Associativity: α · (β · γ) = (α · β) · γ when defined, that is for (α, β), (β, γ) ∈ G ⋆ G,
which implies that (α · β, γ), (α, β · γ) ∈ G ⋆ G.
• Identity: σ(ε(x)) = x = τ (ε(x)), while α · ε(x) = α when x = σ(α), and ε(y) · α = α
when y = τ (α).
• Inverses: each α ∈ G has a two-sided inverse α−1 ∈ G such that σ(α) = τ (α−1 ),
τ (α) = σ(α−1 ), and α−1 · α = ε(σ(α)), α · α−1 = ε(τ (α)).
Remark : For the more abstract-minded reader, a streamlined version of this definition
is to say that a groupoid is “a small category such that every morphism has an inverse”.
See the survey paper [5] and text [25] for further details, references, and applications in
algebra, geometry, topology, crystallography, and elsewhere.
Example 2.3. Let G be a Lie group acting on a manifold M . The action groupoid
is the principal bundle G = G × M , with source and target maps
σ(g, z) = z,
τ (g, z) = g · z
for
(g, z) ∈ G × M.
(2.4)
The groupoid multiplication, inversion, and identity are explicitly given by
(h, g · z) · (g, z) = (h · g, z),
(g, z)−1 = (g −1 , g · z),
ε(z) = (e, z),
(2.5)
for g, h ∈ G, z ∈ S, and e the identity element of G. The verification of the groupoid
axioms is straightforward.
Definition 2.4. Given a Lie group G acting on a manifold M , the symmetry groupoid
of a subset S ⊂ M is the following subgroup of the action groupoid:
GS = { α = (g, z) | z ∈ S, g ∈ Gz } ⊂ G × S,
(2.6)
Thus, the induced source and target maps, multiplication, inversion, and identity on
GS are given by the same formulas (2.4), (2.5). Clearly, if g ∈ Gz then g −1 ∈ Gg·z ; further,
if h ∈ Gg·z , then h · g ∈ Gz , which thus confirms the groupoid structure of GS . Set
Gz = σ −1 { z } = Gz × { z }
to be the source fiber of z ∈ S.
5
(2.7)
Remark : A groupoid is a Lie groupoid if G and M are smooth, meaning C∞ , manifolds, the source and target maps are smooth surjective submersions, and the identity and
multiplication maps are smooth. The action groupoid is trivially a Lie groupoid. However, as we will see in later examples, symmetry groupoids, even of smooth submanifolds
S ⊂ M , are not necessarily Lie groupoids.
Remark : Some authors relax the global symmetry condition (2.1) by only requiring
that g · S ⊂ S. Under this definition, the set of global symmetries only forms a semigroup
in general, since the inverse of g may fail to satisfy the relaxed symmetry condition. For
example, consider the three-parameter semi-direct product group G = R+ ⋉ R 2 that acts
on M = R 2 by translations and dilations:
(x, y) 7−→ (λ x + a, λ y + b)
for
λ ∈ R+ ,
(a, b) ∈ R 2 .
Then, for the line segment
S = { (x, 0) | −1 < x < 1 } ,
(2.8)
say, the set of group transformations g ∈ G satisfying g · S ⊂ S is the scaling semigroup
{ (λ, 0, 0) | 0 < λ ≤ 1 } ( R+ .
Similarly, relaxing the local symmetry condition (2.3) to g · (S ∩ U ) ⊂ S will produce
a “semi-groupoid” of local symmetries. We will not pursue this further extension of the
theory here.
The groupoid elements that fix a point z ∈ S form a bona fide group, G∗z ⊂ Gz ,
satisfying
G∗z × {z} = { α ∈ G | σ(α) = z = τ (α) } ⊂ Gz ,
(2.9)
and known in the groupoid literature as the vertex group of z, [25]. However, I would
prefer to call it the isotropy group or, when seeking to emphasize that these are not global
symmetries, the local isotropy group of the point z, in keeping with standard transformation
group terminology, and also in view of our later use of the term “vertex” in the context of
curve geometry, [15]. Note that if α, β ∈ G have the same source, z = σ(α) = σ(β), and
target, τ (α) = τ (β) then γ = β −1 · α ∈ G∗z . Moreover, the local isotropy groups form a
“normal system” within G in the sense that if g ∈ Gz and h ∈ G∗z , then g · h · g −1 ∈ Gg·z .
Or, to state this in another way, adapting the standard notation for the adjoint map of a
group, [28],
Ad α · G∗σ(α) = G∗τ (α) for any α ∈ G.
(2.10)
Example 2.5. Let G = SE(2) = SO(2)⋉R 2 be the special Euclidean group consisting
of all orientation-preserving rigid planar motions — translations and rotations† — acting
Extending this discussion to the full Euclidean group G = E(2) = O(2)⋉R 2 , that also includes
reflections, is a worthwhile exercise for the reader.
†
6
on the plane M = R 2 . We parametrize G by g = (θ, a, b), for θ ∈ SO(2) ≃ S 1 , (a, b) ∈ R 2 ,
representing the rigid motion
(x, y) 7−→ (x cos θ − y sin θ + a, x sin θ + y cos θ + b).
(2.11)
Let us determine the Euclidean symmetry groupoid for a few basic subsets S ⊂ R 2 .
If S is a circle, then its symmetry groupoid is simply the Cartesian product (principal
bundle) GS = GS × S where GS ≃ SO(2) ⊂ SE(2) is its global symmetry group — the
one-parameter subgroup of rotations around the center of the circle. For example, the unit
circle
S1 = {x2 + y 2 = 1},
has (special) Euclidean symmetry groupoid
GS1 = (θ, 0, 0; x, y) 0 ≤ θ < 2 π, x2 + y 2 = 1 ≃ SO(2) × S1 ⊂ SE(2) × S1 .
The source and target maps on GS1 are
σ(θ, 0, 0; x, y) = (x, y),
τ (θ, 0, 0; x, y) = (x cos θ − y sin θ, x sin θ + y cos θ),
with consequent groupoid multiplication
(ϕ, 0, 0; x cos θ − y sin θ, x sin θ + y cos θ) · (θ, 0, 0; x, y) = (ϕ + θ mod 2 π, 0, 0; x, y), (2.12)
which is only defined when the target of the right hand element matches the source of the
left hand element:
τ (θ, 0, 0; x, y) = σ(ϕ, 0, 0; x cos θ − y sin θ, x sin θ + y cos θ)
= (x cos θ − y sin θ, x sin θ + y cos θ).
If S is only a circular arc, then it has no global special Euclidean symmetries other
than the identity. On the other hand, its symmetry groupoid is not trivial. For example,
the semicircle
S+ = (x, y) x2 + y 2 = 1, x > 0 ,
has symmetry groupoid
n
π
y
πo
GS+ = (θ, 0, 0; x, y) x2 + y 2 = 1, x > 0, − < θ + tan−1 <
( SO(2) × S,
2
x
2
with the same groupoid multiplication rule (2.12). The symmetry groupoid of a closed
circular arc is of the same form at the interior points, but the endpoints only have the
trivial identity map as a local symmetry. (Keep in mind that we are only considering
orientation-preserving Euclidean symmetries here.)
If S ⊂ R 2 is a straight line of infinite extent, its symmetry groupoid is GS = GS × S,
where GS ≃ Z2 ⋉ R ⊂ SE(2) is its global symmetry group consisting of translations in the
direction of the line and 180◦ rotations centered at any point thereon. A bounded open line
segment S — for example (2.8) — has only two global special Euclidean symmetries: the
identity and the 180◦ rotation around its center. On the other hand, the local Euclidean
symmetry set Gz of a point z ∈ S is generated by all translations in the direction of S that
7
map z to another point on S, combined with the 180◦ rotation centered at z; the latter
forms (along with the identity) the isotropy symmetry group G∗z ≃ Z2 of a point z ∈ S.
The case when S is a square is more interesting. The local symmetry set of one of
the corner points is the four element subgroup, Gz ≃ Z4 ⊂ SE(2), containing all rotations
by multiples of 12 π around the center of the square. Indeed, as noted in the introduction,
this is the global special Euclidean symmetry group of the square: GS = Z4 . On the other
hand, if z ∈ S is not a corner, its local symmetry set Gz consists of those translations
that map z to another point on the same side of the square, as in the case of the line
segment, and, in addition, the transformations obtained by following such a translation
by a rotation in the global symmetry group Z4 and/or by a 180◦ rotation centered at the
image point.
Definition 2.6. The symmetry orbit of z ∈ S is the image of source fiber over z
under the target map:
Oz = τ (Gz ) = τ ◦ σ −1 {z} = { g · z | g ∈ Gz } .
(2.13)
By the preceding remarks, there is an evident one-to-one correspondence between the
orbit through z and the quotient of its local symmetry set by its local isotropy group:
Oz ≃ Gz /G∗z . We use the orbits to define an equivalence relation on S, with z ∼ zb if and
only if they belong to the same orbit, or, equivalently, zb = g · z for some g ∈ Gz . Note
that this is well-defined because if zb = g · z for g ∈ Gz , then z = g −1 · zb with g −1 ∈ Gẑ ;
equivalently, zb ∈ Oz if and only if z ∈ Oẑ . Moreover, if ze = h · zb with h ∈ Gẑ , then
ze = (h · g) · z and h · g ∈ Gz . The set of equivalence classes is the symmetry orbit space
or symmetry moduli space of S, and denoted S G = S/ ∼. We let π G : S → S G denote the
projection that maps a point z ∈ M to the equivalence class defined by its orbit Oz .
In Example 2.5, the circle, open circular arc, line, and open line segment each consist
of a single orbit. The square has two orbits, one containing the four corners and the other
consisting of all the remaining points. The tiling in Figure 1, where S consists of all the
line segments and vertices in the figure, i.e., the tile grout† , has four orbits: the points on
the interior of any edge; the interior vertices along with the concave corner vertex where
four edges meet; the side vertices where three edges meet; and the remaining 5 convex
corner vertices where two orthogonal edges meet. The reader is encouraged to determine
the local symmetry sets in each case.
Finally, we make the trivial observation that two globally equivalent subsets have
isomorphic symmetry groupoids.
Proposition 2.7. If Se = h · S are congruent under a group element h ∈ G, then so
are their symmetry groupoids:
GSe = Ad h · GS ≡ β = (h · g · h−1 , h · z) α = (g, z) ∈ GS .
†
Weinstein, [ 39 ], discusses a similar example in detail, but also includes the interior points of
the tiles; however this requires treating the tiling as something other than a single subset of the
plane.
8
3.
Signature.
We now direct our attention to the important case when S ⊂ M is a smooth† —
meaning C∞ — embedded, connected submanifold. We set m = dim M , and 1 ≤ p =
dim S < m. The following definition‡ will prove useful throughout.
Definition 3.1. A piece of the submanifold S is a connected subset Sb ⊂ S whose
interior, under the induced topology of S, is a non-empty submanifold of the same dimension p = dim Sb = dim S, and whose boundary ∂ Sb is a piecewise smooth submanifold of
dimension p − 1. A piece can contain some or all of its boundary points.
For example, a piece of a curve is a connected sub-curve, possibly containing one or
both of its endpoints.
Cartan’s solution to the local equivalence problem for submanifolds under Lie group
and Lie pseudo-group actions relies on the functional dependencies or syzygies among their
differential invariants, [9, 14, 29]. Following [6, 13, 29], we employ a suitable finite collection of differential invariants I1 , . . . , Il to parametrize the differential invariant signature §
Σ = ΣS of the submanifold S. Having prescribed the signature invariants, we define the
associated signature map
χ: S −→ Σ = χ(S) ⊂ Rn ,
χ(z) = (I1 (z), . . . , Il (z)),
(3.1)
whose image is the signature set of S. The determination of an appropriate set of signature
invariants follows from either the exterior differential systems approach to the equivalence
method, [14, 29], or, alternatively, the recurrence formulae for the normalized differential
invariants resulting from the calculus of equivariant moving frames, [13]. The key requirement is that the syzygies among the signature invariants serve to uniquely prescribe the
syzygies among all the differential invariants; examples appear below and in the indicated
references. The selection and overall number of signature invariants required for this purpose depends upon the group G as well as the dimension and, possibly, other intrinsic
properties of the submanifold S.
Example 3.2. Consider the preceding action (2.11) of the special Euclidean group
SE(2) on plane curves S ⊂ M = R 2 . According to [6, 13], the Euclidean signature of a
plane curve is the set Σ ⊂ R 2 parametrized by the Euclidean curvature invariant κ and its
derivative with respect to the Euclidean arc length element ds:
χ: S −→ Σ ⊂ R 2 ,
χ(z) = (κ(z), κs (z)).
(3.2)
The Euclidean signature degenerates to a single point if and only if κ = c is constant, and
so κs = 0, which is equivalent to the curve being an arc of a circle of radius 1/c, or, when
†
The smoothness assumption can evidently be weakened in much of what follows.
‡
I could not find a suitable term in the standard literature. The word “piece” is inspired by
“puzzle piece”, which is an important application, [ 18 ].
§
In [ 13, 29 ], the signature is called the classifying set of S. The term signature was adopted
later in light of significant applications in image processing, [ 6 ], and is now consistently used in
the literature.
9
c = 0, a straight line segment. Otherwise, at least away from singularities and degeneracies,
the signature map traces out an immersed plane curve, typically with self-intersections.
In this case, a complete system of differential invariants is provided by the successive
derivatives of the curvature invariant with respect to arc length: κ, κs , κss , . . . . Assuming
κs 6= 0, the signature curve (3.2) locally determines the syzygy
κs = H(κ)
(3.3)
relating the two signature invariants. Successive differentiation of this syzygy produces the
corresponding syzygies among all the higher order differential invariants; for example, by
the chain rule,
d
d
κss =
κs =
H(κ) = H ′ (κ) κs = H ′ (κ) H(κ),
ds
ds
and so on. This justifies our choice of κ, κs as the signature invariants.
In Euclidean curve geometry, [15], a vertex is defined as a point at which the curvature
κ has an extremal — maximum or minimum — and hence where κs = 0; inflection points of
curvature are usually excluded. In [17], the notion of a generalized vertex was introduced,
and characterized as a point or connected piece of the curve on which κs ≡ 0, the latter
being either a circular arc or straight line segment. Each generalized vertex maps to a
single point χ0 = (κ0 , 0) of Σ, lying on the κ axis, where κ0 equals the reciprocal of the
(signed) radius of the circular arc, or, in the case of a line segment, 0. This implies that
one cannot use the curve traced by the signature map to determine the overall length
of such vertices. This observation was exploited by Musso and Nicolodi, [27], in their
construction of inequivalent closed curves possessing a common Euclidean signature; see
also [16]. In [17], an extension of the standard Euclidean signature that would overcome
these difficulties was proposed, and then exploited in the design of a rather successful
algorithm for automated jigsaw puzzle assembly, [18].
More generally, consider the action of a Lie group G on M = R 2 . We assume that
the action is “ordinary”, meaning that G acts transitively and its prolonged actions on
the curve jet space do not “pseudo-stabilize”. Almost all transitive planar group actions
are ordinary; see [29] for a complete list, based on Lie’s classification tables, [23]. In this
situation, for the induced action of G on plane curves, there is a unique, up to functions
thereof, differential invariant κ of lowest order, known as the G-invariant curvature, along
with a unique, up to scalar multiple, invariant differential form ω = ds of lowest order,
known as the G-invariant arc length element † . The corresponding G-invariant signature
of a curve C ⊂ M takes an identical form, (3.2), parametrized by the curvature invariant κ
and its derivative κs with respect to the G-invariant arc length. Here, a generalized vertex
V ⊂ S is either a stationary point of the curvature invariant, or a maximal connected subset
upon which κ ≡ κ0 is constant, and hence κs ≡ 0. As a consequence of the discussion
following Theorem 3.6 below, this implies that the latter is, in fact, a piece of an orbit of
a suitable one-parameter subgroup H ⊂ G; in other words V ⊂ H · z for any z ∈ V . (See
†
More correctly, ds is the contact-invariant horizontal component of a fully G-invariant oneform on the jet space, [ 13 ]. See Section 4 below for further comments.
10
[33] for a general method for determining the value of the curvature invariant κ0 along
such an orbit.)
Remark : A one-parameter subgroup H ⊂ G is “suitable” when its orbits have a onedimensional global symmetry group, whose connected component containing the identity is
H itself. Orbits that have global symmetry groups of dimension > 1, which hence act nonfreely thereon, are called totally singular , and do not admit well-defined signatures. A wellknown example is when G = SA(2) is the equi-affine group consisting of area-preserving
affine transformations of M = R 2 , [7, 15]. The orbits of one-parameter subgroups H ⊂
SA(2) are conic sections, which, when nonsingular, are the curves of constant equi-affine
curvature κ = κ0 . However, straight lines admit a three-parameter equi-affine symmetry
group, and hence are totally singular. Indeed, the equi-affine curvature is not defined at
inflection points or on straight line segments. See [30] for details, including a Lie algebraic
characterization of the totally singular orbits.
Typically, as in the case of equi-affine plane curves, the differential invariants are not
defined on all submanifolds of the given dimension, and their common domain is a dense
open subset of the submanifold jet bundle. For simplicity, from here on we will assume
that the signature map (3.1) is well-defined at all points of the submanifolds S under consideration. There are extensions of the theory, involving higher order differential invariants
and the like, that can be employed to extend the range of validity of the signature method.
Alternatively, one may be able to avoid singularities where the differential invariants blow
up by assuming that the signature map takes values in a suitably projectivized version of
Euclidean space. However, this approach has yet to be explored in any depth.
Example 3.3. Consider the standard action of the special Euclidean group SE(3) =
SO(3)⋉R 3 on M = R 3 by orientation-preserving rigid motions: translations and rotations.
The basic differential invariants of a smooth curve S ⊂ M are its curvature, which must
be nonzero† κ 6= 0, and torsion τ and their successive derivatives with respect to the
Euclidean-invariant arc length element ds, namely κs , τs , κss , τss , . . . . A signature map
can be constructed from the first three of these, so that
χ: S −→ Σ ⊂ R 3 ,
χ(z) = (κ(z), κs (z), τ (z)).
(3.4)
Observe that we do not need to include τs since, assuming κs 6= 0, i.e., we are not at a
vertex, we can locally express κs = F (κ), and τ = H(κ), which, by the chain rule, uniquely
determines the syzygy
d
H(κ) = H ′ (κ) κs = H ′ (κ) F (κ).
ds
As in the case of plane curves, the higher order syzygies can all be obtained by repeated
differentiation, justifying the above choice of signature invariants.
τs =
†
This assumption is needed in order that the indicated complete set of differential invariants
be well defined. In contrast, straight lines, with κ ≡ 0, are totally singular orbits since they admit
a two-parameter Euclidean symmetry group.
11
Similarly, to analyze the equivalence of two-dimensional surfaces S ⊂ R 3 under the
same Euclidean group action, there are two familiar second order differential invariants:
the Gauss curvature K and the mean curvature H, being, respectively, the product and
average of the two principal curvatures κ1 , κ2 . Again, one can produce an infinite collection
of higher order differential invariants by invariantly differentiating the Gauss and mean curvature. Specifically, at a non-umbilic point where κ1 6= κ2 , there exist two non-commuting
invariant differential operators† D1 , D2 , that effectively differentiate in the direction of the
diagonalizing Darboux frame. A complete system of differential invariants is provided by
K, H, D1 K, D2 K, D1 H, D2 H, D12 K, D1 D2 K, D2 D1 K, D22 K, D12 H, . . . .
For a generic surface, one can use those of order ≤ 1 to parametrize a signature:
χ: S −→ Σ ⊂ R 6 ,
χ(z) = (K(z), H(z), D1 K(z), D2 K(z), D1 H(z), D2 H(z)). (3.5)
The exceptional surfaces are those for which the mean and Gauss curvatures are functionally dependent and yet there is a second functionally independent differential invariant
among their first order invariant derivatives; a complete signature in these cases requires
second order differential invariants. See [29] for further details.
Alternatively, a non-umbilic surface is called mean curvature degenerate if, for any
z0 ∈ S, there exist scalar functions F1 (t), F2 (t), such that
D1 H = F1 (H),
D2 H = F2 (H),
(3.6)
at all points z ∈ S in a small neighborhood of z0 . According to a slight refinement of the
result in [32] — see [36] for details — if S is mean curvature non-degenerate, then one can,
in fact, express its Gauss curvature as a universal rational function of the mean curvature
and its invariant derivatives of order ≤ 4. In other words, for such submanifolds, the mean
curvature H serves to generate the entire algebra of Euclidean surface invariants, and one
can employ it and its derivatives‡
χ
e: S −→ Σ ⊂ R 5 ,
χ
e(z) = (H(z), D1 H(z), D2 H(z), D12 H(z), D22 H(z)).
(3.7)
to parametrize an alternative Euclidean differential invariant signature.
Returning to the general situation, since differential invariants are, by definition, invariant under the (prolonged) action of G, the signature map (3.1) is not affected by the
local symmetries of S. In other words, if g ∈ Gz is a local symmetry based at z ∈ S, then
χ(g · z) = χ(z),
whenever
α = (g, z) ∈ GS .
†
(3.8)
These are not the same as the operators of covariant differentiation, but are closely related,
[ 15, 29 ].
‡
One can show that the mixed derivatives D1 D2 H, D2 D1 H are not needed here.
12
This implies that the signature map is constant on the symmetry groupoid orbits, and
hence factors through the symmetry moduli space:
χ
✲ Σ
❃
❩
✚
✚χ
π G ❩⑦
❩ ✚ e
SG
S
with
χ
e(π G (z)) = χ
e(Oz ) = χ(z).
(3.9)
Let us define the signature rank , or rank for short, of a point z ∈ S to be the rank of
the signature map at z, i.e., the rank of its Jacobian matrix or, equivalently, its differential:
0 ≤ rz = rank dχ|z ≤ p = dim S.
(3.10)
For example, in the case of the Euclidean signature of a plane curve (3.2), the signature
rank is 0 if both κs = κss = 0; otherwise it is 1. We note that rz is a bounded lower
semicontinuous function of z ∈ S, meaning that rz ≤ rẑ for all zb ∈ S in a sufficiently
small neighborhood of z. Equation (3.8) implies that the signature rank is constant on the
symmetry orbits of S. A point z ∈ S is called regular if the signature rank is constant in a
neighborhood of z, i.e., on S ∩ U where U ⊂ M is an open subset containing z ∈ S ∩ U .
Clearly, if z ∈ S is regular, then any point in its symmetry orbit zb ∈ Oz is also regular.
Proposition 3.4. If z ∈ S is regular of rank k, then the image of the signature map
restricted to a suitable neighborhood z ∈ U forms a k-dimensional submanifold χ(S ∩ U ) ⊂
Σ.
Remark : In general, [29], the differential invariant rank of a point z ∈ S is defined as
the maximum, over all n ≥ 0, of the rank of the map defined by all its differential invariants
of order ≤ n. Clearly the signature rank is bounded from above by the differential invariant
rank although generically, away from singularities, they agree. If we knew in advance
the order at which the differential invariant rank is achieved, then we could extend the
signature map by including all the differential invariants up to that order and thus replace
signature rank by differential invariant rank. However, (a) the maximum may not be
achieved until high order, and (b) for most submanifolds, the resulting extended signature
is highly redundant and of scant practical value. For these reasons, we will restrict our
attention to the signature rank from here on.
With the proper choice of signature invariants, the Cartan Equivalence Theorem,
[13, 29], states that the resulting signature completely determines the symmetry orbit of
a regular point. More specifically, two points lie on the same symmetry orbit if and only
if their signatures locally coincide.
Theorem 3.5. If z ∈ S is regular, then zb = g · z ∈ Oz for g ∈ Gz if and only if there
b ⊂ M such that χ(S ∩ U ) = χ(S ∩ U
b ).
exist neighborhoods z ∈ U ⊂ M and zb ∈ U
Note that it is not sufficient to require χ(z) = χ(b
z ) in order that z and zb lie in the same
symmetry orbit. For example, the Euclidean signature of a plane curve (3.2) is typically
an immersed plane curve with self-intersections. If χ(z) = χ(b
z ) ∈ Σ is an intersection
13
point, then the condition in the Equivalence Theorem 3.5 will not hold when the parts of
S near z and zb map to different branches of the self-intersecting signature curve.
Let Sreg ⊂ S be the open dense subset containing all regular points. We decompose
Sreg =
p
[
Sk ,
(3.11)
k=0
where Sk ⊂ Sreg is the set of regular points of rank k. Note that points on its boundary,
z ∈ ∂Sk , are not regular, and, by lower semicontinuity, have rank 0 ≤ rz ≤ k; further, if
rz = k, then z ∈ ∂Sl for some l > k. In particular, all points on ∂Sp have rank strictly
less than p = dim S. Let Σreg = χ(Sreg ) be the corresponding regular part of the signature
set, which we also decompose by setting Σk = χ(Sk ). Since rank χ = k on Sk , we conclude
that Σk ⊂ Rn is an immersed k-dimensional submanifold. We further decompose
Sk =
[
Sk,i
i
into a disjoint union of connected pieces Sk,i ⊂ S, whose images Σk,i = χ(Sk,i ) ⊂ Σ are
connected subsets of the signature.
Another consequence of the Cartan Equivalence Theorem is that, at regular points,
the signature rank also prescribes the size of the local symmetry set, [13, 29].
Theorem 3.6. A point z ∈ Sk is regular of rank 0 ≤ k ≤ p if and only if its local
symmetry set Gz is a local Lie group of dimension p − k, or, more precisely, its connected
e ⊂ G , is a relatively open subset of a (p − k)component containing the identity, e ∈ G
z
z
e
b
b the completion of the
dimensional Lie subgroup, Gz ⊂ Gz ⊂ G. We call this subgroup G
z
local symmetry set Gz .
In particular, the maximally symmetric component S0 ⊂ S consists of regular points
of signature rank 0 (if any). Each of its pieces S0,i maps to a single point of the signature
b
Σ and, in fact, is a piece of an orbit of a suitable p-dimensional subgroup G
0,i ⊂ G.
b
b
Indeed, G0,i = Gz is the completion of the local symmetry set of any point z ∈ S0,i . More
generally, [29]:
Theorem 3.7. If S is connected and of constant rank k, then its local symmetry
b ⊂ G as their
sets all have a common (p − k)-dimensional connected Lie subgroup G
S
b =G
b for all z ∈ S. Moreover, S is the disjoint union of a k-parameter
completion: G
z
S
b . The connected component of its symmetry groupoid,
family of pieces of orbits of G
S
e
b ,
GS∗ = (g, z) z ∈ S, g ∈ G
⊂S×G
(3.12)
z
S
b containing the identity section, and
is an open subbundle of the principal bundle S × G
S
hence a Lie groupoid.
For example, a surface S ⊂ R 3 contained in a non-cylindrical surface of revolution has
constant Euclidean rank 1, and is the disjoint union of circular arcs centered on a common
14
b ⊂ SE(3) of
axis, each contained in a circular orbit of the one-parameter subgroup G
S
rotations around the axis.
More generally, if S is of variable rank, containing more than one nonempty component
Sk ⊂ S consisting of regular points of rank k, then GS is not a Lie groupoid since, according
to Theorem 3.6, its source fiber dimension is not constant on all of S. An example would
be a Euclidean plane curve of the type considered in [27], which contains several pieces of
rank 1 along with some circular arcs and/or straight line segments of rank 0. According
to Theorem 3.7, each connected component Sk,i ⊂ Sk is the disjoint union of pieces of
b ⊂ G,
a k-parameter family of orbits of a common (p − k)-dimensional Lie subgroup G
k,i
b
b
which can be identified as the completion Gk,i = Gz of the local symmetry set of any point
z ∈ Sk,i therein. Of course, the subgroup may well vary from component to component.
In particular, points z ∈ Sp belonging to the component of maximal rank (if such
exists) have a purely discrete local symmetry set Gz ⊂ G. In [6, 29], the number of local
symmetries, # Gz , was characterized by the signature index , which, roughly, counts the
number of points on the original submanifold mapping to the same point in the signature.
More rigorously, let us call a point z ∈ Sp completely regular if Σ is, locally, an embedded
p-dimensional submanifold in a neighborhood of the image point χ(z) ∈ Σ. Thus, complete
regularity excludes points of self-intersection and other singularities of the signature. Let
Sp∗ ⊂ Sp denote the open dense subset of all completely regular points of rank p. The
index of z ∈ Sp∗ is then defined as the number of points in Sp∗ that are mapped to the same
signature point ζ = χ(z):
ind z = # zb ∈ Sp∗ χ(b
z ) = χ(z) = ζ = # χ−1 {ζ} = ind ζ.
(3.13)
Complete regularity implies that the signatures near any point z ∈ χ−1 {ζ} are locally
identical.
The index determines the number of discrete local symmetries of z that do not fix it,
in the following sense. Recall that G∗z ⊂ Gz denotes the local isotropy group of the point z,
that is, all the local symmetries that fix z. The quotient set Gz /G∗z must be discrete, since
it is in one-to-one correspondence with the pieces in S that are equivalent to a suitably
small piece containing z. Theorem 3.5 then implies the following:
Proposition 3.8. The index of a completely regular point z ∈ Sp∗ is equal to the
cardinality of its quotient local symmetry set: ind z = # (Gz /G∗z ).
Remark : The index is not necessarily constant on Sp , even when it is connected, since
the number of local symmetries can vary as one traverses the submanifold. For example,
Sp could contain j rigidly equivalent pieces Sb1 , . . . , Sbj , whose points would have index j,
as well as k 6= j different rigidly equivalent pieces Se1 , . . . , Sek , whose points are of index k.
Example 3.9. In [27], Musso and Nicolodi construct examples of “cogwheels”,
which are closed plane curves that have identical Euclidean signatures, are everywhere
of rank 1 and index n, and yet possess different discrete global symmetry groups GS ⊂
SO(2). Roughly, the construction starts with a closed curve obtained by smoothly joining
n suitable rigidly equivalent curve pieces, each of which is individually of index 1, implying
15
b
b
a
a
b
b
b
a
a
a
a
a
a
a
b
a
b
b
a
b
b
a
b
b
Cogwheels.
Figure 2.
that there are no rigid motions, other than the identity, in its symmetry groupoid. Think
of a regular n–gon whose sides have been symmetrically deformed so that its corners have
also been smoothed out. The resulting curve is of rank 1 and signature index n, and,
moreover, has global symmetry group Zn ≃ { θ = 2 k π/n | 0 ≤ k < n } ⊂ SO(2) consisting
of all rotations through multiples of 2 π/n around its center.
One then splits, in the same manner, each constituent piece into a pair of rigidly
inequivalent subpieces. By suitably rearranging the resulting 2 n curve pieces, one can
produce a closed curve, that they call a cogwheel, that has the same signature and index
as the original, but whose symmetry group is only Zm where m is a given divisor of n.
Figure 2 sketches two examples. The left hand cogwheel has Z6 symmetry, consisting of 12
pieces of two inequivalent types, labelled a and b, respectively; the short transverse lines
mark the ends of the individual pieces. The right hand cogwheel has the same 12 pieces,
rearranged so that it has only Z2 as its global symmetry group. The more subtle aspects of
the construction ensure that the cogwheels remain closed and sufficiently smooth in order
that their signatures be well-defined.
Observe that, while the global symmetry group of such rearranged cogwheels can vary,
their local symmetry set at each point (except possibly at the endpoints where the pieces
have been rejoined) remains of cardinality n and equal to the index. In other words, the
cogwheels all have (essentially) the same index, the same number of local symmetries, away
from the joins, but non-isomorphic symmetry groupoids, which thus serves to characterize
their global inequivalence, cf. Proposition 2.7.
One final remark: the open curve obtained by deleting one of the original pieces of the
cogwheel has no global symmetry (save the identity of course) but this only eliminates one
of the local symmetries at each point, leaving a residual local symmetry set of cardinality
n − 1, and a consequential reduction of its symmetry groupoid.
16
Let us now extend the notion of index to include points which have non-discrete
symmetry, as well as points in Sp that are not completely regular.
Definition 3.10. The index of a regular point z ∈ Sk is defined as the maximal
number of connected components of χ−1 [χ(Sk ∩ U )] where U ⊂ M is a sufficiently small
open neighborhood of z such that Sk ∩ U is connected.
This definition allows us to generalize Proposition 3.8 as follows.
Theorem 3.11. If z ∈ Sreg , the number of connected components of the quotient
Gz /G∗z is equal to its index, denoted ind z.
Theorem 3.12. If Σk = χ(Sk ) is an embedded submanifold, of dimension k =
rank Sk , then there is a one-to-one correspondence between it and the corresponding component of the symmetry moduli space: Σk ←→ SkG = Sk / ∼ ⊂ S G .
4.
Weighted Submanifolds and Weighted Signatures.
In practical applications to image processing and elsewhere, one typically approximates the signature by discretizing (sampling) the original submanifold, and then employing suitable† numerical approximations to compute the signature differential invariants,
the net effect being an approximate discretization of the corresponding signature, [6]. To
maintain accuracy along the entire curve, the sample points should be more or less uniformly distributed, which, in the Euclidean curve case, means that the sample points are
fairly evenly spaced with respect to the arc length. Alternatively, one can employ a random discretization of the curve, as in [4], based on the probability measure determined
by rescaling arc length by the total length of the curve. In the ensuing discussion, we will
ignore numerical inaccuracies, and focus our analysis solely on the effect that uniformly
sampling the submanifold has on its signature.
Remark : We do not discuss the practicalities of constructing such uniform samplings
of submanifolds. Even the “elementary” case of a sphere under surface area measure leads
to the generalized Fekete problem, which appears as one of Smale’s celebrated 18 problems
for the twenty-first century, [38]. Powerful new algorithms for distributing points on
general curved surfaces based on graph Laplacians have been recently developed in [24].
Let us focus our attention initially on the case of curves under the Euclidean group.
After uniform sampling, the number of points in any constituent curve piece is approximately (or probabilistically) proportional to its length. On the other hand, the corresponding points in the signature curve will be distributed according to the measure induced by
pushing forward the arc length measure of the original curve via the signature map. This
remains valid even on a generalized vertex — circular arc or straight line segment —
†
In [ 6, 2 ], the use of symmetry-preserving numerical algorithms for this purpose is advocated
and used. See [ 31, 19, 37, 1 ] for further developments of this approach, based on the calculus of
equivariant moving frames, [ 13 ], leading to important new algorithms for symmetry-preserving
numerical schemes for integration of ordinary and partial differential equations.
17
since the number of points clustering at the corresponding signature point (κ0 , 0) will be
proportional to its overall length. In the limit, as the discretization becomes more and
more dense, the sample point distribution converges, in the usual sense, to the arc length
measure dµ = ds on the original curve, and to the push-forward of arc length measure
dν = χ# dµ on the signature, defined so that if Γ ⊂ Σ, then
Z
−1
ν(Γ) = µ(χ (Γ)) =
ds.
(4.1)
χ−1 (Γ)
If we parametrize the signature Σ by κ, which can be done locally provided κs 6= 0, and
hence away from the problematic rank 0 points corresponding to generalized vertices, then
the measure on Σ is
dκ
dκ
dν = χ# (ds) = ind ζ
= ind ζ
,
ζ = (κ, κs ) ∈ Σ,
(4.2)
| κs |
| H(κ) |
where ind ζ denotes the index of the signature point ζ, and reflects the fact that χ−1 (Γ)
may contain multiple disjoint yet equivalent pieces of the original curve C, while κs = H(κ)
is the local characterization of the signature curve as a graph, cf. (3.3). The absolute value
comes from the fact that, as s increases, in accordance with the chosen orientation on C,
the signature curve is traversed in the left-to-right orientation when κs > 0, but in the
opposite orientation when κs < 0. We conclude that the weight on the signature curve
can be used to determine the index, and hence the number of local symmetries, through
formula (4.2).
On the other hand, at the vertices, where κs = 0, the push-forward measure ν acquires an atomistic term concentrated at the signature point ζ0 = (κ0 , 0), whose weight
is proportional to the total Euclidean length of the circular arc(s) or line segment(s) that
map to the point ζ0 . Thus, the induced weighted measure on Σ is given by
Z
X
dκ
+
ℓ(ζ0 ),
(4.3)
ν(Γ) =
ind ζ
| κs |
Γ
ζ0 ∈Γ ∩ {κs =0}
whenever Γ ⊂ Σ. Here, the atomistic weight
Z
ℓ(ζ0 ) =
ds
(4.4)
χ−1 {ζ0 }
equals the sum of the lengths of all the circular arc/line segments mapping to ζ0 = (κ0 , 0),
whose cardinality equals their common index. However, this has the implication that
the weighted signature does not, in general, uniquely determine the original curve up to
equivalence, since the weight (4.4) at any point ζ0 = (κ0 , 0) ∈ Σ only measures the total
length of all the circular arcs of radius 1/κ0 (or straight line segments when κ0 = 0), and
not the number thereof nor how their individual lengths are apportioned.
Turning to the general scenario, let us first introduce some notation for measure
concentrated on weighted submanifolds. The simplest is the atomic or delta measure
concentrated at a point a ∈ R n , denoted by
1,
a ∈ D,
δa (D) =
for
D ⊂ Rn.
(4.5)
0,
a 6∈ D,
18
Definition 4.1. A weighted submanifold of dimension p is a pair S = (S, | ω |) in
which S is a smooth p-dimensional manifold and | ω | ≥ 0 a non-negative p-density on S.
Remark : We use p-densities, [22], rather than p-forms in our characterization for
two reasons: (a) so that we do not have to worry about orientability of S, and, more
importantly, (b) when we integrate the density on its underlying submanifold, we obtain a
non-negative number — assuming the integral converges. One can, in many places, weaken
the underlying smoothness assumptions, but this is not necessary for the basic applications
we present in the sequel.
Given a weighted submanifold S = (S, | ω |) with S ⊂ R n , we define the corresponding
measure δS concentrated on S by
Z
δS (D) =
|ω|
for
D ⊂ R n.
(4.6)
S ∩D
In particular, if S = (a, w) is a single point a ∈ R n with weight w ≥ 0, then δS = w δa .
More generally, if h(z) ≥ 0 is a measurable function of z ∈ R n , then
h δS = δS̃ ,
where
e = (S, h | ω | )
S
(4.7)
is a reweighted version of S.
Now, let S ⊂ M be a p-dimensional submanifold. Let Ω be a G-contact-invariant†
horizontal p-form on the submanifold jet space Jn (M, p), for suitable n ≥ 0. For example,
one might set
Ω = ω1 ∧ · · · ∧ ωp,
(4.8)
where ω 1 , . . . , ω p form the contact-invariant horizontal coframe constructed by applying
the equivariant moving frame invariantization process to the basic horizontal coframe:
ω i = ι(dxi ), cf. [13]. Another option is to set
Ω = dI1 ∧ · · · ∧ dIp ,
(4.9)
where I1 , . . . , Ip are functionally independent differential invariants. The most general
contact-invariant p-form has the form
Ω = J ω1 ∧ · · · ∧ ωp,
(4.10)
where J is an arbitrary differential invariant, [29]. This added flexibility in the choice of
p-form can play an important role in practical applications. For instance, in the case of
Euclidean plane curves, we could employ an invariantly re-weighted version of arc length,
†
The term contact-invariant refers to the fact that, under the prolonged group action on
the submanifold jet space, Ω is invariant modulo contact forms. This holds even in the case of
Euclidean plane curves, where the arc length form Ω = ds is, in fact, only a contact-invariant
one-form on jet space under prolonged Euclidean transformations. Any contact-invariant form can
be made fully invariant by the addition of a suitable contact correction. In the present situation,
the contact forms play no role — indeed, their defining property is that they vanish on jets of
submanifolds — and so can be safely ignored. See [ 13, 20, 29 ] for complete details.
19
Ω = J ds, where J is a suitable differential invariant, that is, a function of the basic
curvature invariants κ, κs , κss , . . . . For example, to emphasize (or de-emphasize) parts
of high curvature, one might choose Ω = φ(κ) ds where φ(·) is a suitably chosen scalar
function.
Specification of the contact-invariant p-form Ω induces an invariant measure on pdimensional submanifolds S ⊂ M . (For simplicity, we will avoid submanifolds whose
jet does not belong to the domain of the p-form in jet space.) In order to preserve nonnegativity, we replace the p-form Ω by the corresponding p-density | Ω |, [22]. This induces
the G-invariant measure
Z
µ(S) =
|Ω|
(4.11)
S
on compact p-dimensional submanifolds, both orientable and non-orientable. Mimicking
(4.1), we introduce the push-forward measure ν = χ# (µ) on the signature:
Z
−1
ν(Γ) = µ(χ (Γ)) =
|Ω|
for
Γ ⊂ Σ.
(4.12)
χ−1 (Γ)
The component Σp = χ(Sp ) of maximal rank, corresponding to those points in S with only
a discrete local symmetry set, can be locally parametrized by p = dim Σp independent
differential invariants, say I1 , . . . , Ip . Thus, using (4.10), we can write
dν = ind ζ
J dI1 ∧ · · · ∧ dIp
,
∂(I1 , . . . , Ip )
∂(ω 1 , . . . , ω p )
ζ = (I1 , . . . , Il ) ∈ Σ,
(4.13)
where the denominator is the determinant of the p × p Jacobian matrix whose entries are
∂Ij /∂ω k = Ij,k , i.e., the corresponding first order derived invariants, which may well be
included in the signature invariants, or, more generally, can be written in terms of them
using the moving frame recurrence formulae, [13]. In particular, if Ω is given in terms of
the invariants by (4.9), then the denominator in (4.13) is equal to 1.
The key considerations already arise in the case of two-dimensional surfaces in threedimensional space, and so we will discuss this situation in some detail.
Example 4.2. As an educational, elementary example, consider the two-dimensional
abelian group G = R 2 acting intransitively on M = R 3 by translation in the two “horizontal” directions:
(x, y, u) 7−→ (x + a, y + b, u),
(a, b) ∈ G.
(4.14)
We consider the induced action of G on surfaces S ⊂ R 3 which, for simplicity, we assume
to be given as the graphs of smooth functions u = f (x, y). In this case, the differential
invariants are simply
u, ux , uy , uxx , , uxy , uyy , . . . ,
and the invariant differential operators the ordinary total derivatives Dx , Dy . The differential invariants of order ≤ 2 can be used to form the signature map
χ(x, y, u(x, y)) = u(x, y), ux(x, y), uy (x, y), uxx(x, y), uxy (x, y), uyy (x, y) ,
(4.15)
20
albeit with some redundancy as discussed below. The associated signature rank of the
surface is the rank of its Jacobian matrix, and so
ux uxx uxy uxxx uxxy uxyy
rank S|z=(x,y,u(x,y)) = rank
.
(4.16)
uy uxy uyy uxxy uxyy uyyy
The basic invariant† horizontal coframe consists of the one-forms ω 1 = dx, ω 2 = dy,
which produce the invariant two-form
Ω = ω 1 ∧ ω 2 = dx ∧ dy.
The corresponding G-invariant measure (4.11) is
Z
µ(S) =
| dx ∧ dy |.
(4.17)
S
Let π: R 3 → R 2 be projection to the x y–plane, so π(x, y, u) = (x, y). Since we are assuming
that the surface S is the graph of a function u(x, y) for (x, y) ∈ D ⊂ R 2 , then π: S → D is
one-to-one and µ(S) = A(D) equals the area of its projection onto the plane.
Uniformly sampling S with respect to the measure (4.17) is equivalent to uniformly
sampling its projection D with respect to planar Lebesgue measure, and then mapping each
sample point (xk , yk ) ∈ D back to the corresponding point (xk , yk , u(xk , yk )) ∈ S. This
induces a sampling of the signature obtained by evaluating the signature map prescribed
by the differential invariants (4.15) at the sample points. As the sample points become
denser and denser, the result converges to the push forward of the flat horizontal measure
(4.17) to the signature.
Let us look at the three basic cases of surfaces of constant rank 0 ≤ k ≤ 2.
Rank 0 : According to Theorem 3.6, a connected surface has rank 0 if and only if its
signature is a single point, Σ = { ζ0 }, if and only if all its differential invariants are constant,
so u = c, ux = uy = · · · = 0, if and only if S is a piece of a horizontal plane: S ⊂ { u = c }
for some constant c, if and only if its local symmetry sets Gz for z ∈ S are two-dimensional,
and hence open neighborhoods of the identity in the full group: 0 ∈ Gz ⊂ R 2 .
In this case, the weight of the signature point equals the area of the projection D =
π(S), as above, which, because S is horizontal, equals the area of S. In other words, the
signature measure ν of a surface of rank 0 is a delta measure concentrated at the signature
point ζ0 weighted by the area of the surface:
ν = A(S) δζ0 = A(D) δζ0 .
(4.18)
Thus, the weighted signature of such a surface will only prescribe its area, and not its
overall shape.
†
In this case, because the group acts projectably, meaning that the transformed independent
variables do not depend upon the dependent variable, the horizontal one-forms are fully invariant
— no contact corrections are needed.
21
Rank 1 : Again, according to Theorem 3.6, a connected surface is everywhere of rank
1 if and only if it has only precisely one functionally independent differential invariant,
namely u — since if u is constant the manifold is of rank 0 — if and only if its local
symmetry sets Gz are one-dimensional, and whose common completion is a one-parameter
b ⊂ G, if and only if S is the union of pieces of orbits of G.
b Since G
b must be a
subgroup G
one-parameter group of translations, it has the form
(x, y, u) 7−→ (x + b t, y − a t, u)
for some constants (a, b) 6= (0, 0). Thus, the surface S is a piece of a non-horizontal ruled
surface, i.e., a union of line segments of the form { (x + b t, y − a t, u) }, not all lying in
a common horizontal plane { u = c }. If the projection D = π(S) is convex, or, more
generally, its intersection D ∩ L with any line of the form
Lc = { a x + b y = c }
(4.19)
S ⊂ { (x, y, u) | u = h(a x + b y) } ,
(4.20)
is connected, then
where the scalar function h is not constant, as otherwise the surface would have rank 0.
Thus, the surface can be viewed as the graph of a traveling wave, [34; §2.1].
The signature Σ of such a rank 1 surface is a curve. Under the above assumption on
D, the line (4.19) forms what we will call a normal cross-section to the projected group
orbits. When restricted to the curve
Cc = { (x, y, u(x, y)) | (x, y) ∈ Lc } ⊂ S,
lying over Lc , the signature map χ: Cc → Σ is locally one-to-one. Globally, the number
of points in Cc mapping to the same point in the signature curve equals its index, as
in Definition 3.10, which also counts the number of connected components in the local
symmetry set of each z ∈ χ−1 (ζ), that is the number of discrete local symmetries at the
point z beyond the given translational symmetry.
Since Cc is a normal cross-section, the area of the projection D = π(S) is equal to
Z
A(D) =
ℓ(Kx,y ) ds,
(4.21)
Lc
where the integrand equals the length of the line segment
Kx,y = { (x + b t, y − a t) } ∩ D
contained in D that passes through (x, y) ∈ Lc . Thus the weighted signature will have
the probability measure given by the push-forward of the weighted arc-length measure
dµ = ℓ(Kx,y ) ds on the signature curve:
ν = χ# (dA) = δΣ
Σ = (Σ, | ω |) = (Σ, χ# (ℓ ds)).
where
(4.22)
Rank 2 : A surface has rank 2 everywhere if and only if, nearby any point, it admits two
functionally independent differential invariants, and hence at most a discrete translational
22
local symmetry set. As above, one of these independent invariants must be u. Assume
that the other is ux , where functional independence means that their Jacobian determinant
does not vanish:
∂(u, ux )
= ux uxy − uy uxx 6= 0.
(4.23)
∂(x, y)
This implies that we can (locally) write
uy = H(u, ux )
(4.24)
for some function H. Differentiation of this syzygy implies
uxy = Hu ux + Hux uxx ,
uyy = Hu uy + Hux uxy = H Hu + Hu Hux ux + Hu2x uxx .
(4.25)
Thus, once we also prescribe the second order syzygy
uxx = K(u, ux ),
(4.26)
we can determine all the other second order syzygies, and hence, by repeated differentiation, all the higher order syzygies too. Thus, parametrization of a reduced differential
invariant signature for rank 2 surfaces of this type requires only the 4 signature invariants
u, ux , uy , uxx . The only rank 2 surfaces not covered are those for which ux = h(u) but uy
is functionally independent of u. These can be analyzed separately, and the details left to
the reader. To cover both cases simultaneously requires using the signature invariants
χ
e(x, y, u(x, y)) = u(x, y), ux(x, y), uy (x, y), uxx(x, y), uyy (x, y)
(4.27)
to parametrize the reduced signature map χ
e: S → Σ ⊂ R 5 .
Remark : The signature syzygy functions (4.24), (4.26) are not arbitrary. Cross differentiation produces the single integrability condition
where
d2x H = dy K,
(4.28)
∂
∂
+ K(u, ux )
,
∂u
∂ux
∂H
∂H
∂
∂
+ ux
+K
,
dy = H(u, ux )
∂u
∂u
∂ux ∂ux
(4.29)
d x = ux
are the derivations (vector fields) representing implicit differentiation with respect to x
and y, respectively. Note that the integrability condition (4.28) is exactly the requirement
that the derivations (4.29) commute:
[ dx , dy ] = 0.
(4.30)
Moreover, the differential invariants u, ux satisfy the functional independence condition
(4.23) if and only if the derivations dx , dy are linearly independent.
23
If we uniformly discretize S, as above, and let the number of points go to ∞, then
this endows the signature Σ with the weighted measure
du ∧ dux
du ∧ dux
,
dν = ind ζ (4.31)
= ind ζ 2
ux uxy − uy uxx ux Hu + ux Hux K − H K where, as usual, ind ζ denotes the index of the signature point ζ, which equals the number
of non-isotropy local symmetries of any point z ∈ S mapping to ζ = χ(z).
In general, a surface S ⊂ R 3 may have variable rank. Suppose, for example, that
S2 ( S is a nonempty open subset of rank 2. Let’s assume, for simplicity, that the
remainder S1 = S \ S2 is of rank 1. Let Σ1 = χ
e(S1 ), Σ2 = χ
e(S2 ), be, respectively, the
5
regular and singular parts of the signature. Thus, Σ2 ⊂ R is a two-dimensional immersed
surface, typically with self-intersections, which is a relatively open, dense subset of Σ,
while Σ1 = ∂Σ2 represents some sort of one-dimensional “singular boundary” of Σ2 . The
weighted signature measure induced by uniformly discretizing S, as above, will merely be
the sum of the corresponding terms (4.22), (4.31) on the appropriate components of Σ.
Example 4.3. Let us next look at the case of surfaces S ⊂ R 3 under the Euclidean
group SE(3), as in Example 3.3. We assume that S is non-umbilic in order that it admits
a well-defined Euclidean signature.
A rank 0 connected surface is contained in the orbit of a suitable two-dimensional
subgroup G∗ ⊂ SE(3). Being non-umbilic implies that S must be a piece of a cylinder,
and so G∗ ≃ SO(2) ⋉ R. (Spheres and planes are totally umbilic, and, moreover, in the
language of [30], are totally singular submanifolds since their respective symmetry groups,
SO(3) and SE(2), have dimension 3 > 2 and hence act non-freely thereon.) Since the mean
and Gauss curvatures are both constant on S, its signature consists of a single point ζ0 .
Indeed, the Gauss curvature is K = 0 while the mean curvature H = 1/(2 R) is one half
the reciprocal of the radius of the cylinder, while all higher order differentiated invariants
vanish. Uniform sampling of S with respect to the Euclidean-invariant surface area measure
implies, in the limit as the number of sample points goes to ∞, that the weight of ζ0 is
given by the surface area A(S). Consequently, the distribution representing the weighted
signature of the connected rank zero surface S is an atomic measure concentrated at the
point ζ0 that is weighted by the area of the surface:
ν = A(S) δζ0 .
(4.32)
Observe that the weighted signature only determines the area and radius of the cylindrical
piece S, and not its overall shape. Consequently, the weighted signature does not uniquely
determine the global geometry of a connected rank 0 surface.
At the other extreme, a surface S of rank 2 has a two-dimensional signature Σ = χ(S).
Generically, the mean and Gauss curvatures are functionally independent, and hence can
be used to parametrize Σ. Thus, according to (4.13), the corresponding weight on Σ
represents the push-forward of the surface area element on S, giving
dH ∧ dK
,
(4.33)
dν = ind ζ D1 H D2 K − D2 H D1 K 24
where D1 , D2 are the invariant differential operators corresponding to the orthonormal
Darboux frame on S, and ind ζ is the index of the signature point ζ ∈ Σ, which counts
the number of distinct, locally equivalent points on the original surface that map to ζ.
Alternatively, assuming the surface is mean curvature non-degenerate — see the discussion
surrounding (3.6) — one can use the mean curvature H and one of its derivatives, say
H1 = D1 H, to parametrize Σ, as in (3.7). The corresponding weight on Σ takes the form
dH
∧
dH
1
,
dν = ind ζ (4.34)
H1 H12 − H2 H11 where H2 = D2 H, H11 = D12 H, H12 = D2 D1 H, etc.
The most interesting case is a rank 1 surface, that possesses a symmetry groupoid with
b ⊂ SE(3) denote
one-dimensional fibers at each point z ∈ S. Assuming S is connected, let G
the completion of the local symmetry set Gz , which is independent of z ∈ S, as implied
b is a one-dimensional subgroup consisting of either translations,
by Theorem 3.7. Thus G
rotations, or screw motions, whose orbits are straight lines, circles, or helices, respectively.
The surface S is thus ruled by the orbits, and can be considered as a piece of the surface
b to a curve C ⊂ R 3 that intersects the orbits of G
b transversally.
obtained by applying G
b
b
More specifically, let S = G · C, which is a surface of translation (a traveling wave), of
rotation (a surface of revolution), or a helical surface. Then S ⊂ Sb is a piece thereof, and
b on S.
C ∩ S can be viewed as a cross-section to the local action of G
Our goal is to compute the surface area of S and thereby deduce the weighted measure
on its one-dimensional signature curve Σ = χ(S). To this end, let us assume that there
b∈b
exists such a cross-section C0 ⊂ S that is orthogonal to the group orbits. Let v
g be the
b which spans the tangent spaces to the orbits.
infinitesimal generator of the action of G,
Orthogonality requires that the tangent vector t to C0 be everywhere orthogonal to the
b |z = 0 for all z ∈ C0 . We will refer to C0 as a normal
infinitesimal generator: t|z · v
cross-section. Normal cross-section curves always exist locally in a neighborhood of any
z ∈ S. Globally, S can be decomposed into a union of pieces each possessing a normal
cross-section curve. Under this assumption, the area of S can then be computed using the
following interesting formula.
b · C be a surface of rank 1, such that C ⊂ S is a normal
Theorem 4.4. Let S ⊂ G
0
0
b ⊂ SE(3). Let
cross-section to the orbits of the one-parameter subgroup G
Z
ℓ(z) = L(Oz ∩ S) =
ds
Oz ∩ S
denote the length of the piece of the orbit Oz through z that is contained in S. Then
Z
A(S) =
ℓ(z(s)) ds.
(4.35)
C0
Remark : Equation (4.35) is reminiscent of the coarea formula of geometric measure
theory, [12, 26]. In anticipation of our subsequent generalization to arbitrary transformation groups, we will refer to it as the Euclidean coarea formula.
25
Proof : We parametrize S by w(s, t) = exp(t v) z(s), where z(s), for s0 ≤ s ≤ s1 , is the
arc length parametrization of C0 , whereby k dz/ds k = 1, and where t0 (z(s)) < t < t1 (z(s))
parametrizes the piece of the group orbit Oz(s) passing through z(s) ∈ C0 . Then
∂w
dz
∂w
(4.36)
,
= exp(t v)∗
= v|exp(t v) z(s) = exp(t v)∗ v|z(s) .
∂s
ds
∂t
Thus,
Z s1 Z t1 (z(s)) ∂w ∂w dz
A(S) =
∂s × ∂t dt ds =
ds × v|z(s) dt ds
s0
t0 (z(s))
s0
t0 (z(s))
!
Z s1 Z t1 (z(s)) Z s1 Z t1 (z(s)) dz ∂w =
ds v|z(s) dt ds =
∂t dt ds
s0
t0 (z(s))
t0 (z(s))
s0
Z
=
ℓ(z(s)) ds.
Z
s1
Z
t1 (z(s))
C0
The second and next-to-last equalities follow from (4.36) and the norm-preserving properties of Euclidean transformations, while the third equality is a consequence of the normality
condition on C0 .
Q.E.D.
Orthogonality of the orbits to C0 is essential for the validity of (4.35). Indeed, the
b is the translation group in the
simplest case is when C0 is a line segment on the x axis, G
direction of the u axis, so that S is a flat planar region of the form { (x, 0, u) | 0 < u < h(x) }
for some scalar positive function h. The orbits Oz for z = (x, 0, 0) ∈ C0 are the line
segments from (x, 0, 0) to (x, 0, h(x)), of length h(x) and hence (4.35) reduces to the
triviality
Z
A(S) =
h(x) dx.
C0
Clearly this formula requires that C0 be orthogonal to the orbits; it is not valid for the skew
translation (x, y, u) 7−→ (x + c t, y, u + t), for c 6= 0, whose orbits are not perpendicular to
the x axis. More generally, the Euclidean coarea formula (4.35) gives a noteworthy formula
for the areas of surfaces of rotation and helicoidal surfaces.
Now consider the weighted signature Σ of the surface S. Since S is assumed to have
rank 1, Σ ⊂ R l is an immersed curve, where l equals the number of Euclidean signature
invariants, which depends upon which version of the Euclidean signature one employs,
e.g., (3.5) or (3.7) or some alternative. The signature map is constant on the orbits of
b ⊂ SE(3), and hence reduces to a locally one-tothe underlying one-parameter subgroup G
one map that identifies the normal cross-section C0 as a covering of the signature curve:
χ: C0 −→
f Σ. The cardinality of the inverse image χ−1 { ζ } ∩ C0 of a point ζ ∈ Σ equals
the index, and counts the number of discrete local symmetries not in the one-parameter
b Since the invariant measure on S is just the surface area element, its push
subgroup G.
forward to the signature curve will produce a measure concentrated on Σ whose weight is
obtained by pushing forward the weight on C0 given by the integrand in (4.35). Thus,
ν = χ# (dA) = δΣ
where
Σ = Σ, χ# (ℓ(z) ds)
(4.37)
26
Keep in mind that, while the normal cross-section to the group orbits is not unique, the
weighted signature is independent of the choice of cross-section.
Thus, the information supplied by the weighted signature curve can be viewed as representing the normal cross-section weighted by the lengths of the intersecting orbits, which
thereby determine the overall area of the surface via the Euclidean coarea formula (4.35).
This is clearly insufficient to uniquely reconstruct the surface up to rigid motion, since
one can rearrange the orbit pieces through the cross-section while preserving their individual lengths, and hence construct a new, inequivalent surface that still has the same
area and the same signature. For example, the Euclidean-inequivalent parabolic surfaces
of translation
S = {u = x2 , −1 ≤ x, y ≤ 1},
Sb = {u = x2 , −1 ≤ x ≤ 1, x − 1 ≤ y ≤ x + 1},
have identical weighted signature curves since they have a common normal cross-section
C0 = { (x, 0, 0) | − 1 ≤ x ≤ 1 }, and the line segments belonging to S and Sb passing
through a common point in C0 have the same overall length. Thus, while the weighted
signature of a rank 1 surface again does not uniquely determine the surface, it does, as
always, prescribe its local geometry.
More generally, suppose we have a surface S ⊂ R 3 of variable rank. We decompose
S = S0 ∪ S1 ∪ S2 ∪ Ssing
where each Sk has rank k and Ssing is the set of irregular points, which forms the boundary of the open dense subset formed by the union of the regular subsets S0 , S1 , S2 . We
decompose each Sk into connected components, and then form the corresponding signature weight according to (4.32), (4.37) and (4.33) or (4.34). The full weighted signature
measure is obtained by combining the various components, making sure to multiply locally
equivalent parts, which have identical signatures, by the corresponding index.
Our final goal is to generalize the preceding coarea formula and resulting weighted
signature measure to group actions on submanifolds of arbitrary dimension. However, for
simplicity, we will continue to restrict our attention to a Lie group G acting on surfaces
S ⊂ M = R 3 , deferring the completely general case to later contributions. Suppose that
Ω is a G-contact invariant 2-form defined on the surface jet bundle Jn (M, 2), while ω is a
G-contact invariant one-form on the curve jet bundle Jn (M, 1). Both can be systematically
constructed through the method of equivariant moving frames, [13, 20]. We interpret the
corresponding densities, | Ω | and | ω | as, respectively, the G-invariant surface area and
arc length elements. Of course, these are not uniquely defined, because either can be
premultiplied by an arbitrary differential invariant (of the appropriate kind).
The case of rank 0 and rank 2 surfaces proceeds in an evidently analogous fashion to
the above Euclidean case, and so the only case of genuine novelty is the case of a rank
b be the one-parameter group generated by the local symmetries of the
1 surface. Let G
b · z of the (connected component) of G
b are
surface, as per Theorem 3.7. The orbits Oz = G
b∈b
prescribed by the flow of its infinitesimal generator 0 6= v
g.
b and ω be as above. Then the arc length of the orbit piece
Proposition 4.5. Let v
b = { exp(t v
b) z | t0 ≤ t ≤ t1 } ⊂ Oz
O
z
27
is given by
b )=
ℓ(O
z
Z
bz
O
b |z i (t1 − t0 ),
| ω | = h ω|z ; v
(4.38)
b and cotangent
where h · ; · i denotes the natural pairing between the tangent bundle T O
z
∗
b
bundle T Oz .
b is everywhere tangent to its orbits. We first note that
Proof : Keep in mind that v
b i is, in fact, constant along the orbit. Indeed, the derivative of
the scalar quantity h ω ; v
b|exp(t v̂) z i h ω|exp(t v̂) z ; v
b with
with respect to t is a sum of two terms, the first involving the Lie derivative of v
b , which trivially vanishes, and the second Lie derivative of ω with respect to
respect to v
b , which vanishes owing to the assumed G-invariance of the arc length.
v
Thus, we can easily compute the arc length integral using the given orbit parametrization:
Z b
b )∗ ω|exp(t v̂) z ℓ(Oz ) =
exp(t v
bz
O
Z t1 b |z i (t1 − t0 ),
b |exp(t v̂) z i dt = h ω|z ; v
=
h ω|exp(t v̂) z ; v
t0
as claimed.
Q.E.D.
Example 4.6. Consider the case of a space curve, parametrized by (x, y(x), z(x)),
under the Euclidean group SE(3). The Euclidean arc length element is
p
(4.39)
ω = ds = 1 + yx2 + zx2 dx.
b)z0 —
In particular, if the curve coincides with a piece of a Euclidean orbit z(t) = exp(t v
a straight line, circle, or helix, whose character depends upon the infinitesimal generator
Note that, on Oz ,
b = ξ ∂x + η ∂y + ζ ∂z ∈ se(3).
0 6= v
yx =
and hence
bi| =
|hω;v
s
yt
η
= ,
xt
ξ
1+
zx =
(4.40)
zt
ζ
= ,
xt
ξ
p
ζ2
η2
bk
ξ 2 + η2 + ζ 2 = k v
+
|
ξ
|
=
ξ2
ξ2
(4.41)
which is, indeed, constant along the orbit.
The first issue is to determine what replaces the Euclidean condition that the curve
intersect the group orbits in a normal direction, given that there is no G-invariant notion
of inner product in general. Here is the proposed generalization of this condition.
28
Definition 4.7. Under the given choice of arc length and surface area forms, a curve
C ⊂ S ⊂ R 3 will be called a normal cross-section provided it forms a cross-section to the
orbits of the one-parameter symmetry group of the rank 1 surface S that is generated by
b ∈ g and, moreover, at each point z ∈ C, satisfies
0 6= v
b ∧ wi = hω;v
bihω;wi
hΩ;v
for all
w ∈ T C|z .
(4.42)
Remark : The normality condition (4.42) defines an underdetermined system of ordinary differential equations whose solutions prescribe the curve C; their local existence can
be established by standard techniques.
Example 4.8. Let us investigate (4.42) in the case of the Euclidean group SE(3).
We use the standard Euclidean arc length element (4.39) and surface area element, which,
assuming S is the graph of u(x, y), is
q
(4.43)
Ω = 1 + u2x + u2y dx ∧ dy.
Let the curve be parametrized by z(τ ) = (x(τ ), y(τ ), u(τ )). Since C ⊂ S, we must have
u(τ ) = u(x(τ ), y(τ ))
hence
uτ = u x x τ + u y y τ .
Similarly, given that the orbits with infinitesimal generator (4.40) must lie in S, the same
computation implies that
ζ = ux ξ + uy η.
Solving the last two equations for ux , uy produces
ux =
η uτ − ζ y τ
,
η xτ − ξ y τ
uy =
ζ x τ − ξ uτ
.
η xτ − ξ y τ
Thus, on C
b
|v
Ω| =
q
ξ dy − η dx = kv
b × zτ k dτ.
(η xτ − ξ yτ )2 + (ζ xτ − ξ uτ )2 + (η uτ − ζ yτ )2 η xτ − ξ y τ On the other hand, in view of formula (4.41), this is equal to the right hand side of (4.42)
if and only if
b × zτ k = k v
b k k zτ k
kv
which demonstrates orthogonality of the tangent to the curve and the tangent to the orbit
under the Euclidean inner product.
With this in hand, we can state a general G-invariant coarea formula.
Theorem 4.9. Let C ⊂ S be a normal cross-section in a rank 1 surface S ⊂ M .
Then
Z
ZZ
b ∩ S) | ω | .
|Ω| =
ℓ(O
(4.44)
z
S
C
b ∩ S) of the piece of the orbit of a
Here, for each z ∈ C, the G-invariant arc length ℓ(O
z
b ∈ g lying in S is given by formula (4.38).
one-parameter subgroup generated by v
29
b ) z(τ ), for
Proof : Let z(τ ) for τ0 ≤ τ ≤ τ1 parametrize C, and hence w(t, τ ) = exp(t v
t0 (τ ) ≤ t ≤ t1 (τ ), parametrizes our rank 1 surface S. Then,
ZZ
|Ω| =
S
=
Z
τ1
τ0
Z
τ1Z t1 (τ )
τ0
t0 (τ )
b ∧ wτ i dt dτ =
hΩ;v
Z
τ1Z t1 (τ )
τ0
b i t0 (τ ) − t1 (τ ) h ω ; wτ i dτ =
hω;v
t0 (τ )
Z τ1
τ0
b i h ω ; wτ i dt dτ
hω;v
b ) h ω ; w i dτ =
ℓ(O
z
τ
Z
C
b ) | ω |,
ℓ(O
z
b i along the orbit, and formula (4.38). Q.E.D.
where we used (4.42), the constancy of h ω ; v
We conclude that the weighted signature of such a rank 1 surface is the distribution
concentrated on the signature curve which, as in the Euclidean case, is in one-to-one
correspondence with the normal cross-section curve C. The weight is given by the pushforward, under the signature map, of the measure on C given by the integrand in the G
coarea formula (4.44). Similar results extend to surfaces of variable rank.
Acknowledgments: I would like to thank to Orn Arnaldsson and Francis Valiquette
for alerting me to several typos in an earlier version.
30
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